510.7 

Sch9m  SCHWATT 

MODERN  TENDENCIES  IN 
THE  TEACHING  OF  MATH 


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MODERN  TENDENCIES  IN  THE 


TEACHING  OF  MATHEMATICS 


By  Professor  ISAAC  J.  SCHWATT 


READ  AT  THE  MEETING  OF  THE 

New  York  Section  of  the  Association  of  Teachers  of  Mathematics 
December  ii,  1908 


Reprinted  from 

Mathematics  Teacher,  published  by  the  Association  of  Teachers  of  Mathematics 
in  the  Middle  States  and  Maryland 


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4 


MATHEMATICS- 
' LIBRARY 


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MODERN  TENDENCIES  IN  THE  TEACHING  OF 
MATHEMATICS. 

By  Professor  Isaac  J.  Schwatt. 

Until  the  middle  of  the  last  century,  mathematics  had  been 
developed  only  from  the  top,  so  to  speak;  but  during  the  last 
decades,  the  efforts  of  some  of  the  ablest  mathematicians  have 
been  directed  towards  obtaining  clearer  conceptions  of  the  foun- 
dations of  mathematics. 

Cantor,  Weierstrass,  Dini,  Dedekind  and  many  others  have 
studied  the  fundamental  conceptions  of  number  and  space,  and 
their  work  has  resulted  in  more  accurate  ideas  about  these 
conceptions. 

With  all  the  advance  in  the  knowledge  of  mathematics,  with 
our  more  thorough  conceptions  of  the  foundations  of  the  sci- 
ence, and  with  the  resulting  tendencies  to  change  the  methods 
of  presenting  its  various  branches,  the  effect  on  the  student’s 
knowledge  of  the  subject  is  far  from  encouraging. 

During  the  last  few  years,  quite  a little  unfavorable  criticism 
has  been  made  on  the  efficiency  of  our  schools.  Many  of  these 
criticisms  refer  to  the  results  we  now  obtain  as  compared  with 
those  obtained  before.  We  do  not  concern  ourselves  with 
the  criticisms  made  by  skeptics ; they  have  been,  and  they  always 
will  be  with  us.  Nor  ought  we  to  take  seriously  the  statement 
by  George  Bernard  Shaw : “ Those  who  can,  do ; and  those  who 
cannot,  teach.”  But  there  have  been  criticisms  made  by  members 
of  our  own  craft.  Professor  Charles  W.  Earned,  of  the  United 


41 


42 


THE  MATHEMATICS  TEACHER. 


States  Military  Academy,  in  an  article  entitled,  “The  Ineffi- 
ciency of  onr  Public  Schools,”  North  American  Revieiv,  Sep- 
tember, 1908,  tells  us  that  out  of  314  candidates  who  submitted 
in  June,  1908,  to  the  examination  for  entering  West  Point, 
154  failed  in  algebra,  and  237  in  geometry.  The  examination 
questions  were  not  difficult.  They  were  free  from  what  are 
often  termed  ‘ catch  questions,’  and  have  impressed  me  as 
being  eminently  fair.  I understand  that  during  the  examina- 
tion the  candidates  are  very  closely  watched,  and  that  they  have 
to  depend  entirely  on  their  own  resources. 

The  results  of  the  examinations  in  other  subjects  are  corre- 
spondingly lamentable.  This  is  the  more  surprising  since,  as  a 
rule,  only  the  ablest  candidates  are  chosen,  and  many  win  their 
appointment  by  competitive  examination.  The  candidates  rep- 
resent a class  of  earnest  men,  206  out  of  the  314  having  either 
wholly  or  in  part  earned  their  own  living  while  at  school  and 
while  preparing  themselves  for  the  Academy. 

In  1905  the  Association  of  Mathematical  Teachers  in  New 
England  addressed  letters  to  teachers  of  physics  and  chemistry 
throughout  the  schools  and  colleges  of  New  England,  inquiring 
as  to  their  students’  ability  to  perform  arithmetical  and  alge- 
braical operations.*  One  of  the  questions  of  this  inquiry  reads : 
“If  you  have  taught  for  some  time,  please  state  whether  you 
notice  any  difference  in  the  pupil’s  preparation  in  mathematics 
now  and  formerly.” 

Some  of  the  answers  would  indicate  that  the  pupils  have 
deteriorated  under  the  modern  methods  of  teaching,  or  what- 
ever the  cause  might  be. 

Professor  Barns,  of  the  Department  of  Physics,  Brown  Uni- 
versity, writes : “ There  is  a change  for  the  worse.  Coming 
from  the  high  school  of  Cincinnati  to  Columbia  College  thirty- 
one  years  ago,  we  were  decidedly  better  prepared  in  mathemati- 
cal subjects  than  the  men  I teach  now  within  forty  miles  from 
the  ‘Hub  of  the  Universe.’” 

Professor  Bartlett,  of  the  Department  of  Chemistry,  Dart- 
mouth College,  reports : “ I refuse  to  teach  arithmetic.  . . . 
Most  of  the  students  are  helpless.  Incorrect  calculations  are  so 
common  that  the  first  attention  is  always  directed  to  the  calcula- 

Third  rc‘i)orl  of  the  Association  of  Mathematical  Teachers  in  New 
Knfj[lan(l,  ](}o6. 


at 


5 \0.7 

CL.V\ 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  43 

tions  in  case  of  erroneous  reports.  . . . Problems  involving  a 
little  algebra  usually  throw  down  the  whole  division  ....  I 
have  been  teaching  here  for  twenty-six  years  and  had  four 
years  experience  in  secondary  schools  previously.  I think  stu- 
dents are  much  less  competent  in  arithmetic  now  than  formerly.” 

As  a member  of  the  teaching  profession,  I feel  keenly  the 
humiliation  and  the  sting  of  the  criticisms  on  the  results  of  our 
efforts.  If  the  president  of  one  of  our  leading  universities  can 
give  expression  to  the  following:  “We  all  know  that  the  chil- 
dren of  the  last  two  decades  in  our  schools  have  not  been 
educated.  With  all  of  our  training,  we  have  trained  nobody. 
With  all  of  our  instructing,  we  have  instructed  nobody” — there 
must  be  something  decidedly  wrong  with  the  teaching  of  the 
subjects  of  the  curriculum  and  among  them  with  the  teaching  of 
mathematics,  which  is  acknowledged  to  be  one  of  its  strongest 
educational  mediums.  I am  sometimes  surprised  at  the  apparent 
indifference  and  complacency  with  which  we  teachers  in  general 
seem  to  take  these  criticisms.  If  the  criticism  on  the  efficiency 
of  our  schools  is  justified,  it  is  our  duty  by  single  and  united 
effort  to  remove  the  cause  of  it.  If  education  is  good  at  all, 
it  must  be  effective,  and  its  results  must  be  commensurate  with 
the  high  importance  we  attach  to  it,  and  the  large  amount  of 
money  we  expend  on  it. 

Whether  the  fault  lies  with  us  teachers,  or  with  those  who 
are  charged  with  the  administration  of  our  schools,  or  with  the 
object  of  our  efforts,  the  pupil,  upon  us  devolves  the  duty  to 
study  the  conditions  earnestly,  carefully  and  persistently,  and  to 
bring  about  such  changes  as  will  improve  the  existing  conditions. 

It  is  pathetic  to  think  that  our  efforts  have  not  been  crowned 
with  better  success.  On  all  other  questions  affecting  the  wel- 
fare of  the  community  and  the  state,  every  one  of  its  mem- 
bers has  his  say.  But  the  average  man  or  woman  pays  very 
little  attention  to  the  work  of  the  teacher.  We  teachers  have 
it  all  our  own  way.  We  can  make  up  our  curriculum  and 
choose  our  methods  of  teaching  without  interference  from 
the  public.  The  public  judges  us  by  the  results  of  our  labors 
only.  It  is  therefore  the  more  our  duty  to  do  our  best  and  to 
see  that  our  efforts  shall  be  most  effective. 

In  a paper  entitled  “ Our  Duty  as  Teachers,”  I have  con- 
sidered one  of  the  causes  for  the  apparent  failure  of  our  efforts. 


9 


44 


THE  MATHEMATICS  TEACHER. 


This  cause  is  the  tendency  of  the  student  to  conceal  his  ignor- 
ance from  his  teacher,  instead  of  frankly  trying  to  obtain  all 
possible  help  in  his  difficulties. 

There  was  a time  when  the  peculiarities  of  the  teacher  were 
of  such  a nature  as  often  to  enable  a pupil,  who  was  not  true 
to  his  own  interests,  to  pass  a subject  on  insufficient  knowledge. 
The  pupil  took  advantage  of  these  peculiarities  of  the  teacher 
who  was  easily  led  to  have  a higher  opinion  of  the  pupil’s  dili- 
gence and  knowledge  than  he  deserved.  P)Ut  the  day  of  such 
teachers  has  passed.  They  survive  only  in  the  comic  papers. 
The  teacher  of  to-day  ought  to  be  well  able  to  cope  with  the 
pupil  who  hides  his  ignorance  intentionally  or  otherwise. 

In  this  connection  I am  reminded  of  a little  story  which  is 
quite  characteristic  of  the  old-time  professor.  It  is  told  of  a 
professor  in  one  of  our  largest  universities,  that  while  walking 
on  the  campus  and  meditating  upon  a problem,  he  met  a student. 
Linking  arms  with  the  student  the  professor  proceeded  to  ex- 
plain the  problem  to  him  and  to  demonstrate  its  solution.  When 
he  was  at  the  end  of  the  demonstration,  he  said  to  the  student, 
“Don’t  you  see  that  the  result  is  (let  Is  say)  a?”  The  stu- 
dent, who  had  not  paid  much  attention  and  was,  in  fact,  unable 
to  follow  the  demonstration  replied  with  surprise,  “Is  it?” 
The  professor,  believing  that  the  student  while  following  the 
demonstration  had  obtained  a different  result,  went  over  the 
demonstration,  and  found  that  his  first  result  was  incorrect.  The 
apparent  knowledge  and  ability  of  the  student  so  impressed  the 
professor  that  from  that  time  on,  the  student  ranked  in  the 
mind  of  the  professor  as  one  of  the  ablest  mathematicians  in 
the  university,  always  receiving  the  highest  marks  in  the  subject. 

The  failure  of  the  pupil  to  derive  from  his  studies  the  mental 
development,  for  which  they  are  intended,  renders  futile,  at 
least  in  part,  the  aims  and  the  purposes  of  education. 

What  then  is  the  purpose  of  sending  children  to  school? 
What,  in  short,  is  the  puprose  of  education?  On  the  answer 
to  tliis  (jucstion  depend  the  subjects  to  be  taught  in  the  school, 
the  metliod  of  instruction,  and  the  entire  attitude  of  the  teacher 
towards  the  student. 

In  answering  tliis  (piestion,  we  shall  exclude  from  our  con- 
sideration the  professional  schools,  the  industrial  and  trade 
schools,  etc.,  and  shall  consider  the  elementary  and  such  sccon- 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  45 


dary  schools  and  colleges  as  give  what  we  term  liberal  courses 
only. 

Now,  as  I understand  it,  the  purpose  and  aim  of  education 
should  be  the  physical,  moral  and  intellectual  development  of  the 
young,  whose  fitness  shall  determine  what  the  world  is  to  be  in 
time  to  come. 

It  is  true  beyond  discussion,  that  in  order  to  do  right,  to  be 
useful  to  ourselves  and  to  our  fellowmen,  we  must  first  and 
above  all  be  healthy  and  strong.  Our  bodies  must  have  the 
power  of  resistance  to  enable  us  to  discharge  the  manifold 
duties  incumbent  upon  us.  '* 

The  school  should  make  provision  for  the  physical  develop- 
ment of  the  child.  The  school  ought  to  teach  the  laws  of 
hygiene  and  the  rules  of  right  living.  The  child  must  be  taught 
that  the  proper  care  of  the  body  is  as  important  as  the  care  of 
the  mind,  that  one  of  the  means  for  keeping  the  body  in  perfect 
health  is  the  use  of  proper  and  well  prepared  foods,  of  pure 
water,  pure  air,  etc. 

The  second  function  of  the  school  ought  to  be  the  formation 
of  character,  the  inculcation  of  moral  habits,  of  habits  of  right 
acting,  and  of  all  that  is  true  and  noble  and  just. 

Most  schools  seem  to  consider  the  intellectual  development  of 
the  youth  as  their  most  important,  if  not  their  only  function ; 
but  as  the  motto  of  the  University  of  Pennsylvania,  “ Literse 
sine-moribus  vanae  ” (learning  without  morals  is  vain),  em- 
phasizes the  fact  that,  one  of  the  most  important  functions  of  the 
school  ought  to  be  the  development  of  character,  the  inculcation 
of  habits  of  sincerity  and  frankness,  of  diligence  and  perse- 
verance, of  honesty  of  purpose  and  application  to  duty,  of  rever- 
ence and  kindness  and  consideration  of  others. 

A person  possessed  of  these  qualities  and  only  moderate  in- 
telligence is  a more  desirable  member  of  society  and  is  bound  to 
succeed  better  in  life,  in  the  real  sense  of  the  word,  than  a 
person  who  is  lacking  in  some  of  them,  no  matter  how  great 
his  intelligence  or  how  strong  his  mental  powers  might  be. 

We  must  take  into  consideration  that  there  are  many  parents 
who  have  not  the  time  to  supervise  the  moral  education  of  their 
children.  There  are  persons  who  have  assumed  the  responsi- 
bilities of  parenthood  without  being  well  fitted  for  this  great 
task.  On  that  account,  very  often  they  do  not  enjoy  the  rever- 


46 


THE  MATHEMATICS  TEACHER. 


ence  of  their  children.  The  latter,  as  a rule,  show  more  respect- 
for  their  teacher  than  for  their  parents.  With  such  children 
especially  lies  the  great  power  of  the  teacher,  and  his  opportuni- 
ties and  possibilities  to  develop  and  influence  character.  More- 
over since  the  child  spends  the  greater  part  of  the  day  in  school 
and  in  the  preparation  of  his  lessons,  there  is  not  much  time  left 
for  even  the  intelligent  parents,  with  high  moral  ideas,  to  con- 
duct the  moral  education  of  their  children.  The  duty  of  mould- 
ing the  character  of  the  child  therefore  devolves,  in  part  at  least, 
upon  the  school  and  the  teacher. 

It  is  perhaps  a fortunate  circumstance  that  in  most  schools 
no  formal  provision  is  made  for  the  moral  education  of  the 
child  and  no  dogmatic  teaching  of  morals  is  included  in  the 
curriculum.  If  without  preaching  to  the  child,  we  use  every 
opportunity  to  draw  a lesson  in  morals  and  character  building, 
the  purpose  will  be  much  better  accomplished. 

We  now  come  to  the  third  function  of  the  school — the  intel- 
lectual development  of  the  young.  We  believe  in  the  goodness 
of  human  nature,  and  in  the  instinctive  desire  of  every  indi- 
vidual to  do  right,  if  he  can  only  think  right.  I am  in  full 
accord  with  Madame  de  Stael,  that  all  bad  actions  are  only  the 
result  of  thoughtlessness.  Mental  development,  power  of  mind, 
the  ability  to  think  right,  are  the  first  requisites  for  character. 
If  we  could  only  think  right  in  all  matters  pertaining  to  our- 
selves and  to  our  relations  with  others,  we  would  also  act  right. 

There  was  never  a time  when  people  were  so  easily  convinced 
by  arguments  without  submitting  them  to  the  careful  scrutiny 
of  tlieir  reason  and  understanding.  People  seem  now  more 
willing  than  ever  to  have  their  thinking  done  for  them.  There 
was  never  a time  when  the  power  of  the  self-constituted  leader 
who  takes  it  upon  himself  to  do  the  thinking  for  the  people,  was 
so  great  as  it  is  to-day,  when  all  kinds  of  sects  of  religion  and  all 
kinds  of  social  theories  were  so  prevalent ; when  the  influence 
of  the  “ walking  delegate,”  whose  office  is  to  hand  out  ready- 
made ideas  to  his  constituents,  helpful  or  baneful,  was  as  strong 
as  it  is  to-day. 

Just  as  we  must  submit  all  our  actions  to  the  test  of  the 
highest  standards  of  morals,  so  we  must  submit  everything  we 
learn,  read,  or  hear  to  the  test  of  our  understanding.  We  very 
often  think  that  we  understand  an  idea,  but  when  we  come  to 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  4/ 


express  it,  or  carry  it  into  execution,  we  find  that  the  idea  is 
not  entirely  clear  to  us. 

One  of  the  most  pernicious  habits  of  the  mind,  against  which 
we  must  constantly  and  most  vigilantly  guard  our  students,  is 
that  of  superficiality;  of  accepting  an  idea  before  submitting 
it  to  the  test  of  his  understanding;  of  being  satisfied  to  take  up 
another  idea  before  the  first  one  is  entirely  clear  and  assimilated. 

This  habit  of  superficiality  is  only  a prelude  to  a slip-shod 
way  of  doing  things,  to  be  satisfied  when  a task  has  been  done 
so  that  the  result  will  look  right,  yet  not  stand  close  scrutiny 
or  careful  examination.  To  do  every  task  right  and  to  the 
best  of  one’s  ability  is  an  important  and  essential  quality  in 
every  occupation  in  life.  The  time  to  acquire  this  all-important 
habit  is  in  childhood,  and  the  place  to  begin  such  a preparation 
is  in  the  classroom,  whatever  the  subject  taught. 

Our  number  system,  which  is  so  well  adapted  to  the  perform- 
ing of  arithmetical  operations  in  a more  or  less  mechanical  way, 
and  in  this  lies  the  claim  for  the  superiority  of  our  decimal  sys- 
tem over  all  other  systems,  has  contributed  a great  deal  towards 
the  pernicious  tendency  of  doing  things  mechanically,  without 
much  exercise  of  the  power  of  mind. 

The  pupil  carries  this  tendency  to  his  work  in  all  the  other 
mathematical  subjects.  Since  the  young  find  concentration  of 
the  mind  the  hardest  task,  they  easily  give  way  and  do  things 
mechanically  wherever  possible. 

While  the  method  of  the  ancients,  like  the  Egyptians  and  the 
Romans,  for  carrying  out  arithmetical  calculations  were  rather 
cumbersome,  one  cannot  help  thinking  that  these  methods  were 
more  conducive  to  the  acquiring  of  mental  power,  than  the  re- 
fined methods  we  apply  in  our  calculations  to-day. 

A child  who  recites  a poem  without  understanding  its  con- 
tents receives  very  little  benefit  from  memorizing  it.  The  per- 
nicious habit  which  the  child  acquires  of  saying  things  without 
understanding  them,  is  a danger  to  its  moral  and  intellectual 
development.  We  must,  above  all,  educate  the  children  to  think 
and  to  reason.  While  thinking  is  the  hardest  work,  we  can  so 
cultivate  this  all-important  habit  of  the  mind  in  the  child  that  it 
will  enjoy  reasoning,  and  find  thinking  an  agreeable  task.  No 
person  can  discharge  his  duties  of  citizenship  unless  he  is  able 
to  reason  correctly,  and  to  think  right. 


48 


THE  MATHEMATICS  TEACHER. 


It  is  the  duty  of  the  teacher  to  see  that  the  student  has  a clear 
understanding  of  every  idea  presented  to  him,  and  that  he 
acquires  the  habit  of  submitting  every  idea  to  the  test  of  his 
reasoning  before  accepting  it  and  calling  it  his  own.  Every  idea 
introduced  must  be  of  such  a nature  that  the  student  can  gain  a 
clear  understanding  of  it,  his  age  and  mental  development  being 
taken  into  consideration. 

It  is  the  duty  of  the  school  and  the  college  to  pay  special 
attention  to  those  students  who  are  less  gifted,  or  who  are  lack- 
ing in  the  sense  of  duty,  and  have  not  the  strength  of  character 
to  apply  themselves  to  their  studies.  If  schools  and  colleges 
drop  young  men  from  their  rolls  because  they  do  not  show 
aptitude  and  inclination  for  work,  or  because  they  are  unruly, 
such  youths  without  our  help  and  encouragement  may  become 
a burden  to  themselves  and  to  those  around  them.  If  they  do 
not  do  their  duty  in  their  work  in  life,  there  are  other  men  and 
women  eager  to  take  up  their  positions,  and  with  little  senti- 
ment they  are  simply  dismissed. 

It  is  the  special  duty  of  the  school  to  improve  such  youths, 
to  strengthen  their  character,  to  build  them  up  mentally,  and  if 
possible,  make  useful  men  and  women  of  them.  In  the  long 
run,  we  shall  pay  the  penalty  if  we  neglect  to  help  those  who 
need  our  help  most. 

The  student  who  leaves  school  or  college  because  he  is  not 
successful  in  his  studies  often  does  well  when  he  engages  in 
manual  work  or  in  an  occupation  which  requires  little  mental 
effort  and  little  continuous  exercise  of  his  power  of  thinking. 
If  the  failure  in  their  studies  is  due  to  lack  of  ability,  it  might 
be  better  for  such  youths  to  continue  their  studies,  and  by 
proper  training  and  direction  acquire  the  power  of  mind,  with- 
out which  their  usefulness  as  citizens  must  necessarily  be 
impaired. 

The  function  of  the  school  is  to  prepare  the  child  and  the 
youth  for  life.  The  pupil  ought  therefore  to  be  taught  that  all 
of  liis  actions,  while  at  work  or  at  play,  must  conform  to  the 
highest  moral  standards.  He  must  understand  that  to  go  to 
school  is  an  occu])ation,  that  it  is  as  important  for  him  to  be 
successful  in  this  occu])ation,  as  for  his  father  in  that  which  he 
])nrsiK‘S,  and  he  will  be  the  better  fitted  for  his  work  in  life  the 
better  he  a])plies  himself  to  all  phases  of  his  school  life. 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS. 


49 


The  satisfaction  resulting  from  having  done  work  right  is 
invigorating  and  is  conducive  to  true  happiness.  Some  of  us 
perhaps  remember  the  worry  we  experienced  when  we  were 
not  sufficiently  prepared  in  our  lessons,  and  the  anxiety  we  have 
felt  lest  we  should  be  called  upon  to  recite.  Those  children 
who  are  diligent  and  have  all  other  good  qualities,  and  are  there- 
fore free  from  anxiety  and  fear  of  punishment,  are  as  a rule  the 
healthiest  and  happiest.  What  is  true  of  adults  is  true  of  them. 
If  we  are  successful  in  our  efforts,  we  are  happy  and  as  a rule 
healthy,  however  hard  we  may  work.  It  is  worry  and  not  work 
that  impairs  our  health.  If  we  all  think  right,  we  shall  act  right, 
and  we  shall  be  less  subject  to  worry,  and  more  endured  to  work. 

All  the  functions  of  the  school  are  so  interrelated  that  if 
one  is  neglected  the  others  are  bound  to  suffer.  To  do  right,  we 
must  think  right,  and  to  think  right  we  must  be  healthy  in  body 
and  mind.  Mens  sana  in  corpora  sano.  To  be  healthy  we 
must  have  an  easy  conscience,  we  must  be  able  to  decide  and 
choose  what  is  good  for  our  mind  and  body. 

To  acquire  knowledge,  the  student  must  possess  certain  traits 
of  character.  He  must  have  the  qualities  of  diligence,  sincerity 
and  perseverance.  If  he  is  lacking  in  these,  it  is  the  duty  of  the 
school  to  develop  them,  not  only  because  they  are  all-important 
for  every  occupation  in  life,  but  because  without  them  his 
mental  development  is  not  possible.  Any  failure  to  awaken  and 
develop  those  qualities  in  the  pupil  and  the  consequent  failure  to 
derive  the  mental  benefits  from  his  studies,  are  bound  to  cripple 
his  character  for  life. 

The  character  of  the  child  which  is  still  passive,  with  no 
expressed  tendency  for  good  or  evil,  can  be  formed  in  such  a 
manner  as  to  acquire  habits  which  will  influence  his  life  for 
good.  If  he  acquires  while  at  school  the  habits  of  diligence, 
of  being  punctual,  of  being  sincere,  of  being  perseverant  in  his 
work,  he  is  most  likely  to  carry  these  habits  to  his  work  in  life. 
If  he  is,  however,  negligent  in  performing  his  duties  while  at 
school,  he  is  likely  to  shirk  his  duties  when  he  is  a man.  If  he 
pretends  to  know  what  he  really  does  not  understand,  he  will 
acquire  the  habit  of  insincerity.  If  he  is  quieting  his  conscience 
and  making  himself  believe  that  he  understands  an  idea  without 
giving  his  understanding  a searching  and  scrutinizing  test,  this 
will  tend  to  create  in  him  the  disastrous  habit  of  superficialty. 


50 


THE  MATHEMATICS  TEACHER. 


If  he  is  inclined  to  be  irregular  in  attendance  while  at  school, 
he  is  likely  to  show  this  same  tendency  later  as  a man.  What  an 
injustice  to  the  child  who  will  have  to  pay  the  ])enalty  for  any 
failure  in  his  character  due  to  the  neglect  of  the  teacher! 

We  must  teach  our  students  that  diligence,  perseverance  and 
sincerity  are  as  desirable  qualities  in  the  youth  as  they  are  in  the 
grown-up  person ; that  there  is  only  one  step  between  school  life 
and  what  we  call  practical  life;  that  since  the  habits  acquired 
in  youth  are  most  lasting,  he  must  acquire  habits  of  right  action, 
and  practice  them  in  all  his  activities  while  at  school. 

The  methods  of  education  ought  to  be  the  same  whether  the 
child  is  to  become  a laborer,  a mechanic,  or  a professional  man. 
The  standard  of  morals  and  the  ability  to  think  right  and  to 
act  right  ought  to  be  the  same  whatever  may  be  the  station  or 
the  occupation  of  a person  in  life.  Just  as  there  is  one  legal 
code,  there  is  only  one  moral  code,  and  we  all  have  to  conform 
to  its  teachings.  Whatever  advantages  outside  of  the  school 
the  child  of  the  well-to-do  may  enjoy  over  the  child  of  the 
poorer  classes,  each  must  have  the  same  advantages  for  his 
physical,  moral  and  mental  development,  if  he  is  to  be  able  to 
make  use  in  the  same  measure  of  the  opportunities,  which  our 
country  offers  to  all  its  citizens. 

The  stability  of  this  republic  depends  on  the  ability  of  the  in- 
dividual to  thing  and  to  act  right  and  on  his  proper  conception  of 
happiness  and  in  what  ways  true  happiness  and  enjoyment  in  life 
can  be  attained. 

The  unsatisfactory  results  of  our  efforts  have  caused  teach- 
ers of  mathematics,  and  especially  associations  like  this  to  look 
for  remedies.  This  dissatisfaction  has  made  itself  felt  in  cer- 
tain tendencies  in  the  teaching  of  our  subject. 

In  the  following  we  shall  discuss  some  of  the  aspects  of  these 
tendencies  and  make  a few  suggestions,  in  the  hope  that  those 
who  do  not  agree  with  the  views  herein  presented,  will  yet 
consider  them  of  sufficient  weight  to  continue  their  discussion, 
and  that  some  conclusions  may  be  reached  which  each  and  every 
one  of  us  will  endeavor  to  apply  for  the  betterment  of  the  teach- 
ing of  our  subject. 

There  is  a demand  for  better  text-books,  more  thorough  and 
more  lucid,  for  syllabi  to  serve  as  guides  in  determining  what 
to  choose  in  the  teaching  of  a subject  and  what  to  omit.  I say 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  5 I 


it  with  a great  deal  of  hesitancy  that  in  most  cases  the  prepara- 
tion of  a syllabus  is  a misdirected  effort,  and  that  the  use  of  a 
syllabus  may  have  an  ill  effect  on  the  true  purposes  of  teaching. 
With  the  increase  in  the  number  and  in  the  variety  of  text- 
books for  all  grades  in  the  same  subject,  a syllabus  can  be  dis- 
pensed with. 

Moreover  I feel  that,  as  a rule,  the  purpose  of  a syllabus  is  to 
serve  as  a guide  as  to  what  topics  some  of  the  colleges  expect 
candidates  for  admission  to  be  most  efficient  in.  But  since  the 
great  majority  of  secondary  school  pupils  do  not  enter  college, 
a syllabus  has  no  place  in  such  schools. 

There  is  a tendency  to  represent  the  subjects  of  the  curriculum, 
including  mathematics,  in  such  a manner  as  to  incite  the  interest 
of  the  learner.  But  if  the  young  are  brought  up  with  the  idea 
that  everything  they  do  must  incite  their  eagerness  or  curiosity, 
there  is  no  wonder  that  those  who  fail  to  secure  work  which  is  in 
accord  with  their  tastes  are  augmenting  the  ranks  of  those  whose 
theories  may  mean  ruin  to  civilization  and  to  the  progress  of 
mankind. 

Those  are  blessed  who  have  found  their  work.  The  large 
majority  of  men  and  women,  however,  must  find  the  inspiration 
for  their  work,  and  for  doing  it  right,  in  the  consciousness  that 
they  are  doing  something  to  help  the  common  weal ; that  it  gives 
them  the  means  to  provide  for  their  own  needs  and  for  the  needs 
of  those  who  are  dependent  on  their  support,  that  if  they  do  their 
duty  to  the  best  of  their  ability,  they  may  in  time  find  work 
which  is  more  congenial  to  them. 

The  children  and  the  youth  ought  to  be  taught  to  do  their 
duty.  The  teacher  should  insist  upon  such  a course  on  the  part 
of  the  student  with  firmness  and  strictness. 

The  methods  by  which  children  are  made  to  acquire  the  habit 
of  doing  their  work  and  doing  it  promptly  and  well,  will  differ 
with  the  individual  child.  _Some  children  will  give  heed  to  a 
word,  others  will  be  influenced  by  persuasive  talk,  again  others 
will  do  right,  only  after  more  or  less  severe  punishment. 

What  an  injustice  to  the  pupil,  and  what  harm  is  done  to  his 
entire  future,  if  the  teacher  does  not  insist  that  he  be  accurate, 
punctual,  and  diligent,  and  that  he  devote  himself  with  all  his 
energy  to  his  work.  The  pupil  when  he  leaves  school  to-day  and 
finds  employment  to-morrow  will  be  judged  by  his  conduct  and 


52 


THE  MATHEMATICS  TEACHER. 


by  his  diligence.  If  he  fails  to  comply  with  the  requirements 
that  govern  his  employment,  he  may  lose  it,  which  may  mean 
humiliation  and  disappointment,  and  may  result  in  sad  con- 
sequences for  him,  and  for  those  nearest  to  him. 

Education  ought  to  be  a preparation  for  life,  as  life  is,  or  as 
it  ought  to  be.  The  pupil  and  the  student  of  to-day  are  the 
employer  and  employe  of  to-morrow,  and  what  they  will  be,  the 
world  will  be. 

The  best  preparation  for  life  is  to  make  the  young  do  that 
which  requires  effort — effort  of  thought,  effort  to  overcome  any 
possible  disinclination  to  work;  to  derive  pleasure  from  doing 
one’s  very  best  and  from  the  satisfaction  of  overcoming  diffi- 
culties. 

Life  is  a continuous  rendering  of  service.  Service  in  order  to 
earn  a livelihood ; service  in  bringing  up  our  children  and  in 
caring  for  those  around  us ; service  in  taking  part  in  the  higher 
purposes  of  life — moral,  social  and  intellectual. 

Wherever  we  turn,  there  are  conditions  to  improve  and  work 
to  be  done,  which  call  for  service,  hard  and  unremitting  service, 
and  very  often  with  only  the  reward  that  comes  from  the  con- 
sciousness of  having  done  something  worth  while. 

There  is  a tendency,  even  in  the  secondary  schools,  to  present 
the  various  subjects  of  mathemathics  with  a view  to  their  possi- 
ble application  in  life,  believing  that  education  in  order  to  be 
effective,  ought  to  be  practical. 

It  is,  however,  a matter  of  experience  that  we  soon  forget 
whatever  we  learn  at  school,  and  it  is  true  beyond  question  that 
in  life,  we  need  more  than  anything  else,  health,  character  and 
power  of  mind,  for  on  these  our  success  will  largely  depend. 
We  can  get  along  with  very  little  mathematics,  very  little 
geography  and  science,  very  little  of  the  other  subjects  in  the 
curriculum,  if  we  have  only  character  and  the  ability  to  think 
right. 

Even  professional  schools  can  give  their  students  only  such 
knowledge  as  forms  the  foundation  of  their  special  science 
and  of  what  they  will  meet  in  the  practice  of  their  professions, 
d'he  school  ought  to  imbue  its  students  with  the  scientific  spirit 
and  with  the  thirst  for  knowledge.  When  they  enter  their  pro- 
fession they  will  have  continually  to  study  and  to  learn.  Any 
change  in  the  hy])()t]iesis  of  their  science,  every  new  discovery 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  53 


and  invention  in  it  they  will  have  to  apply,  if  they  do  their  full 
duty. 

The  great  majority  of  men  and  women  will  have  little  use  in 
life  for  mathematics,  beyond  the  operations  with  integers  and 
simple  fractions.  It  is  greatly  to  be  regretted  that  even  in  the 
engineering  profession,  mathematics  is  being  used  less  and  less, 
in  the  actual  solving  of  practical  problems.  Some  of  the  larger 
manufacturing  establishments,  such  as  the  Cambria  Steel  Com- 
pany and  the  Carnegie  Steel  Company  are  publishing  handbooks 
for  engineers  which  contain  formulae  and  all  kinds  of  data 
which  an  engineer  is  likely  to  need  in  his  work.  These  books 
have  created  a class  of  engineers  called  “ handbook  engineers  ” 
who  use  the  formulae  and  the  tabulated  data  very  often  without 
a knowledge  and  understanding  of  the  methods  by  which  they 
have  been  obtained.  The  use  of  such  books  tends  to  lessen  the 
mental  activity  of  those  who  use  them. 

A large  engineering  company  has  lately  provided  its  engineer 
with  slide  rules,  expecting  them  to  be  used  in  making  computa- 
tions, and  thus  to  save  time. 

But  just  as  it  is  impossible  to  prepare  the  student  of  engineer- 
ing in  all  problems  which  he  may  meet  in  practice,  so  it  is  im- 
possible to  include  in  a handbook  all  the  data  he  may  need  in 
his  profession.  In  some  cases  he  will  have  to  fall  back  on  his 
mathematical  knowledge  and  the  more  thorough  it  is  the  better 
fitted  he  will  be  for  the  work  requiring  mathematical  theories, 
the  results  of  which  are  not  to  be  found  in  a handbook.  The 
world  is  a vast  store  of  undiscovered  energies  and  opportunities. 
We  must  only  sharpen  our  minds  to  discover  them,  and  acquire 
the  knowledge  and  ability  to  exploit  them  for  the  benefit  and 
comfort  of  mankind. 

The  greater  usefulness  of  the  college-trained  engineer  com- 
pared with  the  engineer  who  has  learned  his  profession  in  the 
factory  and  in  the  field,  the  so-called  practical  engineer,  lies  in 
•the  higher  development  of  power  of  mind  in  the  former  and  in 
his  ability  to  use  such  methods  and  results  as  his  knowledge  of 
mathematics  and  mechanics  affords. 

In  manufacturing  the  aim  is  to  produce  an  article  in  the  short- 
est possible  time,  with  the  least  expenditure  of  material,  but 
with  due  regard  to  quality  and  efficiency.  Now  the  practical 
experience  and  his  natural  instinct  will  often  tell  the  engineer 


54 


THE  MATHEMATICS  TEACHER. 


in  what  way  a saving  of  material,  etc.,  can  be  effected,  but  in 
many  cases,  such  a knowledge  can  be  obtained  by  mathematical 
theory  only. 

Take  for  instance,  the  manufacture  of  reservoirs,  of  basins, 
or  simply  of  tin  cans.  We  can  construct  tin  cans  of  different 
shapes,  and  therefore  from  different  amounts  of  material,  but 
all  of  one  and  the  same  content,  a quart,  say.  Now,  unless  there 
is  a reason  that  a can  should  be  of  a certain  shape  (as  a can  used 
for  baking  powder,  or  for  spices,  etc.)  it  ought  to  be  constructed, 
for  a given  volume,  with  the  least  amount  of  material,  i.  e.,  with 
the  view  to  economy.  Mathematical  calculations  lead  to  the 
result  that  to  construct  a can  having  a top  and  bottom,  the  least 
amount  of  material  will  be  required  if  the  diameter  of  the  base 
is  equal  to  the  height  of  the  can,  and  the  larger  the  diameter  of 
the  base  of  a cylinder  with  neither  a top  nor  a bottom,  the  less 
material  will  be  necessary  to  construct  it. 

As  a result  of  the  tendency  to  make  the  work  in  arithmetic 
more  interesting  and  practical,  the  child  often  learns  how  to  use 
a commutation  ticket  or  a mileage  book,  and  the  names  of  parts 
of  cattle  in  the  language  of  the  butcher,  but  does  not  get  the 
mental  training  which  can  be  obtained  from  a proper  study  of 
the  subject,  nor  the  ability  to  perform  with  accuracy  some  of  its 
simplest  operations.  Such  a course  is  as  much  to  be  con- 
demned as  the  tendency  of  some  people  to  provide  themselves 
with  luxuries,  without  being  able  to  satisfy  the  more  substan- 
tial necessities  for  their  body  and  mind. 

There  are  so  many  faculties  and  abilities  to  be  developed  in 
the  young,  that  the  school  cannot  accomplish  even  the  smallest 
part  of  them.  It  would  certainly  be  most  desirable  to  teach  the 
children  to  model,  and  to  paint,  and  to  play  a musical  instru- 
ment, and  to  sing. 

It  would  be  highly  desirable  for  the  school  to  discover  any 
latent  talents  and  abilities  of  the  pupil.  Some  persons  have  not 
been  aware  of  possessing  high  talents,  which  would  have  re- 
mained latent  if  they  had  not  been  discovered  by  accident.  But 
until  the  school  can  fulfill  well  the  great  and  difficult  tasks  of 
developing  the  body,  the  character,  and  the  mind  of  the  young, 
no  other  functions  should  he  included  in  its  work. 

ddie  scliools  cannot  teach  all  that  is  worth  knowing,  all  that 
we  ought  to  know  in  order  to  understand  social  and  economic 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  55 


conditions  and  changes,  the  workings  of  nature  and  to  enjoy 
its  beauties.  But  the  school  ought  to  awaken  the  desire  for 
knowledge.  It  should  train  the  young  in  power  of  observation, 
in  power  of  discrimination,  power  of  concentration,  and  in  all 
those  abilities  and  qualities  of  the  mind  which  are  developed  by 
the  study  of  the  different  branches  of  the  school  curriculum, 
and  in  no  small  measure  by  the  study  of  mathematics. 

There  is  a tendency  to  correlate  the  different  branches  of 
elementary  mathematics,  especially  algebra,  with  some  of  the 
branches  of  the  natural  sciences,  particularly  with  physics,  and 
mechanics.  There  are  a number  of  causes  which  have  led  to  this 
tendency,  one  of  them,  I feel,  has  as  its  base  the  dssatisfaction 
of  teachers  of  physics  and  engineering  with  the  average  stu- 
dent’s ability  to  perform  with  facility  and  accuracy  algebraical 
and  arithmetical  operations,  and  to  solve  simple  equations  or 
the  different  letters  which  occur  in  them.  Also,  the  equations 
in  most  of  the  books  on  algebra  have  integral  coefficients,  and 
give  for  the  value  of  the  unknown,  integers  or  simple  fractions, 
but  many  of  the  problems  in  physics  and  engineering  lead  to 
equations  whose  coefficients  are  decimals  and  the  results  are 
very  often  complicated  fractions  which  must  be  expressed  in 
decimals.  I claim  that  the  student  who  is  well  trained  in  per- 
forming arithmetical  and  algebraical  operations  and  who  has  a 
clear  conception  of  the  principles  underlying  these  operations, 
ought  to  be  able  to  handle  algebraical  expressions,  and  to  solve 
the  equations  to  which  many  of  the  problems  in  physics  or 
engineering  may  lead. 

There  are  three  kinds  of  problems  from  physics  and  mechanics 
that  might  be  used  to  illustrate  mathematical  operations.  To 
understand  the  conditions  of  one  kind  of  these  problems,  a 
thorough  knowledge  of  the  principles  of  the  physical  and  me- 
chanical phenomena  involved  is  necessary,  but  to  effect  their 
solution  only  the  simplest  arithmetical  operations  need  be  used. 
The  solution  of  such  problems  belongs  in  a course  of  physics. 
The  age  and  mental  development  of  the  pupil  when  he  studies 
arithmetic  will  as  a rule  not  permit  of  the  thorough  understand- 
ing of  the  principles  of  physics  and  mechanics  necessary  to 
translate  the  conditions  of  these  problems  into  the  language  of 
arithmetic. 

There  are  problems  from  physics  and  mechanics  in  the  solu- 


56 


THE  MATHEMATICS  TEACHER. 


tion  of  which  the  methods  of  algebra  may  be  used  to  advantage, 
and  in  which  the  translation  of  the  given  conditions  leads  to 
very  simple  algebraical  expressions.  The  solution  of  this  class 
of  problems  also  belongs  more  properly  in  a course  in  physics. 
By  introducing  into  such  problems  conditions  which  are  arti- 
ficial, the  translation  of  the  conditions  and  the  resulting  alge- 
braical expressions  may  be  increased  to  any  degree  of  difficulty. 

The  great  majority  of  problems  from  physics  and  engineer- 
ing, some  even  of  the  simplest  kind,  need  for  their  solution  a 
knowledge  of  calculus,  and  many  of  them,  a knowledge  of  the 
more  advanced  branches  of  mathematics,  like  differential  equa- 
tions, definite  integrals,  etc. 

To  solve  these  problems,  perfect  facility  in  handling  alge- 
braical and  trigonometrical  expressions,  as  well  as  a thorough 
knowledge  of  the  conceptions  and  operations  of  calculus  is 
necessary. 

I feel  that  the  correlation  between  mathematical  subjects 
and  those  of  some  of  the  natural  and  applied  sciences  has  at 
its  base  also  the  desire  of  the  teacher  to  satisfy  the  tendency  to 
make  the  study  of  mathematics  practical  and  interesting  to  the 
student.  But  most  of  the  problems  from  physics  and  mechanics 
which  I have  seen  or  have  been  able  to  compose,  and  the  solution 
of  which  can  be  effected  by  algebraical  operations,  are  of 
little  practical  use.  There  is  not  much  more  practical  infor- 
mation in  figuring  out  how  long  it  takes  a boat  to  cross  a 
river  under  certain  conditions  of  wind  and  current,  than  there 
is  in  figuring  out  how  old  Ann  is.  To  calculate  how  long  it 
would  take  a boat  to  actually  cross  a river,  it  would  be  neces- 
sary first  to  determine  the  velocity  of  the  wind,  the  velocity  of 
the  current,  the  weight  of  the  boat  and  so  on.  To  obtain  some 
of  these  data,  a knowledge  of  meteorology,  hydrodynamics,  etc., 
and  skill  in  handling  delicate  apparatus  are  necessary. 

The  entire  question  as  it  appears  to  me  is  whether  the  formal 
study  of  mathematics,  especially  of  algebra,  has  its  place  in  the 
curriculum,  and  whether  the  formal  study  of  this  subject  helps 
to  develop  qualities  of  mind  which  are  desirable. 

Just  as  it  recjuircs  different  ingredients  to  make  up  the  diet 
for  the  body  to  kee|)  all  the  organisms  healthy  and  strong,  and 
in  a haniKmions  state  of  dcvel()])nicnt,  so  the  mind  needs  for 
its  development,  a curriculum  embracing  different  subjects. 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  5/ 


The  study  of  formal  algebra,  without  any  regard  to  its  appli- 
cation, teaches  continuous  concentration  of  attention,  uncon- 
scious following  of  a number  of  rules  in  more  or  less  rapid 
succession,  and  is  an  excellent  discipline  for  acquiring  habits 
of  precision,  accuracy  and  power  of  mind. 

Facility  in  handling  algebraical  expressions  is  a necessary 
preparation  for  all  other  branches  of  mathematics.  Without  a 
thorough  knowledge  of  the  principles  of  algebra  and  perfect 
facility  in  performing  arithmetical  and  algebraical  operations, 
the  student  will  not  be  able  to  gain  a knowledge  of  the  prin- 
ciples of  trigonometry,  solid  geometry,  calculus  and  other 
branches  of  mathematics,  and  to  solve  the  problems  of  these 
subjects.  The  student  will  need  all  the  time  which  is  ordinarily 
given  in  the  secondary  schools  to  the  study  of  algebra,  to  acquire 
facility  in  this  subject,  and  not  until  he  has  attained  perfect 
facility  in  performing  algebraical  operations,  should  other  mat- 
ter be  introduced. 

No  student  can  hope  to  acquire  a thorough  mastery  of  a 
branch  of  mathematics  unless  he  is  able  to  perform  with  little 
effort  such  operations  from  other  branches  as  may  be  needed 
in  this  one. 

When  he  takes  up  the  study  of  algebra,  the  student  must  be 
able  to  perform  with  ease  such  arithmetical  operations  as  he 
may  need  in  algebraic  work.  In  the  study  of  trigonometry  he 
must  have  perfect  facility  in  such  algebraical  and  geometrical 
work  as  may  be  required  in  the  solution  of  the  trigonometrical 
work.  Again,  if  he  is  to  successfully  acquire  a thorough  knowl- 
edge of  the  principles  and  operations  of  calculus,  he  must  have 
perfect  mastery  of  such  arithmetical,  algebraical,  geometrical 
and  trigonometrical  work  as  he  may  need  in  the  work  in 
calculus. 

Just  as  when  we  have  acquired  facility  in  spelling  and  writing, 
we  need  pay  little  attention  to  these,  but  many  concentrate  our 
efforts  on  the  formulation  of  our  ideas. 

Unless  the  results  of  a problem  are  accurate,  they  are  of 
little  use.  To  carry  a mathematical  operation  through  all  of 
its  stages  to  an  accurate  result,  requires  a thorough  knowl- 
edge of  the  subject  and  a facility  which  can  only  be  acquired  by 
constant  practice  and  by  hard  work.  It  is  a great  injustice  to 
the  student  who  chooses  a profession  like  engineering,  to  use 


58 


THE  MATHEMATICS  TEACHER. 


the  time  needed  to  acquire  facility  in  performing  algebraical 
operations,  in  the  solutions  of  problems  to  arouse  his  interest, 
or  which  appear  to  him  to  he  practical. 

The  educational  value  and  the  mental  development  which  is 
derived  from  the  study  of  physics  is  in  a sense  different  from 
the  educational  benefit  to  he  derived  from  the  study  of  mathe- 
matics. There  can  be  no  objection  to  the  use  of  occasional 
illustrations  from  physics  or  mechanics,  or  from  any  of  the 
sciences,  to  show  the  student  the  power  of  mathematical  study, 
outside  of  its  value  as  an  educational  medium,  but  the  student 
must  have  a thorough  knowledge  of  the  principles  and  laws  of 
the  sciences  from  which  the  illustrations  are  drawn  in  order 
to  translate  intelligently  the  conditions  of  such  problems  in  the 
language  of  mathematics. 

In  the  teaching  of  mathematics,  we  must  be  very  careful  not 
to  give  way  to  fads  or  to  apply  new  methods  and  tendencies, 
unless  we  are  sure  that  they  will  lead  to  more  effective  results. 

We  must  not  experiment  on  the  child  that  has  but  a single 
opportunity  to  gain  or  to  lose.  In  the  interest  of  the  children 
who  are  to  make  the  world  what  it  will  be  during  the  next 
generation,  and  in  a way,  during  all  generations  to  come ; in  the 
interest  of  the  larger  purposes  of  true  education,  I wish  to 
sound  a warning  that  we  exercise,  as  far  as  the  teaching  of 
mathematics  is  concerned,  the  greatest  care  and  caution  before 
applying  a “modern  tendency’’  to  the  minds  of  the  young.  For 
years  we  have  been  following  in  some  of  the  elementary  branches 
of  mathematics,  the  German  methods  of  teaching.  There  is 
now  a decided  tendency  to  follow  Professor  John  Perry,  of  the 
Royal  College  of  Sciences  of  London,  who  terms  the  German 
method  the  “ intellectual  Strassburg  goose-stufhng,”  and  in  an- 
other place,  the  “ intellect  by  order  of  the  authorities.”  He  calls 
the  English  system  “ antiquated,”  and  has  a method  all  his  own.* 

This  country  has  proven  the  most  fertile  soil  for  all  kinds  of 
experiments  in  the  teaching  of  mathematics.  We  have  been 
quick  to  adopt  methods,  especially  if  advocated  by  foreign 
])edagogues. 

Our  course  in  this  respect  has  been  similar  to  that  of  a person 
who  attaches  a higher  appreciation  to  a thing,  even  if  of  in- 

* School  Mai/icmdtics,  Marcli,  1004,  p.  195. 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  59 


ferior  make  and  quality,  simply  because  the  label  states  that 
it  is  made  in  a foreign  country. 

The  Grube  system,  long  after  it  has  been  discarded  in  the 
land  where  it  originated,  is  still  in  use  in  many  schools  here. 
While  the  method  of  teaching  and  the  selection  of  text-books 
are  in  some  foreign  countries  in  part  regulated  by  the  Govern- 
ment, and  therefore  more  or  less  uniform,  it  is  left  here,  as  a 
rule,  to  every  city,  and  in  some  places  to  every  school,  to  choose 
its  text-books  and  the  method  of  presenting  the  subject. 

It  is  true  that  we  are  a young  nation,  but  we  have  already 
reached  maturity.  We  ought  not  to  follow  the  utilitarian  ideas 
in  education  of  one  nation,  nor  the  impractical  ones  of  another. 
We  ought  to  develop  a system  of  teaching  mathematics — if  we 
find  the  present  system  wanting — which  is  peculiarly  our  own, 
which  is  in  accordance  with  our  ideals  and  inspirations,  our 
temperament  and  environments,  in  short,  which  is  American. 

While  Germany  was,  and  in  a measure  still  is,  the  nursery  for 
the  world  in  some  of  the  sciences  and  arts,  other  nations  of 
Europe  have  used  the  German  ideas  only  to  reach  their  own 
maturity.  They  have  then  developed  their  sciences  and  arts 
in  accordance  with  their  own  national  tendencies  and  peculi- 
arities. 

The  Italians  have  developed  a distinct  school  of  music,  and 
also  of  some  of  the  sciences — among  them  mathematics. 

The  French  have  a distinct  school  of  the  art  of  painting,  and 
have  methods  and  tendencies  in  some  of  the  sciences  which 
are  peculiar  to  them  and  to  their  national  characteristics. 

No  matter  how  experienced  the  teacher  may  be,  he  must  be 
sure  that  he  has  a thorough  mastery  of  an  idea  before  present- 
ing it  to  his  students,  and  he  must  plan  and  lay  out  in  every 
detail  the  work  which  he  is  to  take  up  with  the  class.  In  sup- 
port of  this  argument,  I may  be  permitted  to  state  an  incident 
of  my  own  experience.  After  having  taught  a certain  subject 
for  a number  of  years,  both  in  the  classroom  and  in  private 
tutoring,  I felt  that  I was  no  longer  in  need  of  going  over  the 
details  of  the  lesson  before  meeting  the  class.  It  happened, 
however,  that  during  one  year  I had  two  sections  in  one  and  the 
same  subject,  on  the  same  days  of  the  week,  but  at  different 
hours.  The  sections  did  not  follow  each  other  on  the  different 
days  in  the  same  order;  on  some  days  one  section  and  on  other 


6o 


THE  MATHEMATICS  TEACHER. 


clays  the  other  section  came  first.  I found  that  after  teaching 
one  section,  I could  do  the  work  with  the  next  section  more 
effectively,  and  go  over  the  ground  in  a shorter  time  than  with 
the  first.  To  give  both  sections  as  equal  advantages  as  possible, 
I made  it  a rule  to  go  over  the  lesson  before  meeting  the  first 
section  just  as  though  I were  a novice  in  teaching  the  subject. 

The  abstruse  manner  in  which  a subject  is  presented  must  not 
be  confused  with  scientific  treatment.  The  more  difficult  a 
mathematical  idea,  the  more  lucidly  and  clearly  it  ought  to  be 
presented  to  save  the  student  time,  strength,  and  often  dis- 
couragement. 

Mathematics  is  advancing  at  such  a rapid  rate  that  it  is  becom- 
ing quite  impossible  even  for  those  who  have  high  ability  and 
much  leisure  to  devote  to  its  study,  to  obtain  a knowledge  of  all 
its  different  branches.  If  this  immense  amount  of  subject  matter 
is  not  presented  lucidly,  as  is  the  case  in  some  of  the  works  of 
the  foremost  writers  on  mathematics,  the  task  of  acquiring 
mathematical  knowledge  is  quite  discouraging,  even  to  the  ablest 
students. 

The  great  masters  in  their  flights  of  genius  perhaps  ought  not 
to  be  expected  to  descend  to  the  level  of  average  mortals  ; but 
the  writers  of  text-books,  and  those  who  record  the  results  of 
their  research  but  who  do  not  belong  to  this  very  exclusive  class, 
ought  to  present  their  ideas  and  their  work  in  a clear  and  lucid 
manner.  The  late  Professor  Tait  must  have  assumed  that  all 
the  readers  of  his  rather  abstruse  “ Elementary  Treatise  on 
Quaternions,”  Cambridge,  1890,  would  bring  to  it  the  same 
ability,  experience  and  knowledge  in  the  subject  which  he  has 
possessed  when  he  wrote  the  book.  The  following  quotation 
from  this  book  page  no,  shows  his  sad  lack  of  sympathy  with 
those  who  are  perhaps,  less  gifted  but  v/ho  are  striving  for 
knowledge.  “ [The  words  above,  ‘it  is  evident,’  have  been  ob- 
jected to  by  more  than  one  correspondent.  But,  on  full  con- 
sideration, I not  only  leave  them  where  they  are,  but  put  them 
in  Italics.  For  they  are,  of  course,  addressed  to  the  reader  only; 
and  it  is  to  be  presumed  that,  before  he  reaches  them,  he  has 
mastered  the  contents  of  at  least  the  previous  sections  which 
bear  on  tliis  (juestion.  If,  with  these  sections  in  his  mind,  he 
does  not  see  tlie  ‘ evidence,’  he  has  begun  the  study  of  qua- 
ternions too  soon.  I ” 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS. 


6l 


There  is  among  teachers  a great  deal  of  difference  of  opinion 
as  to  the  method  of  conducting  the  work  in  the  classroom, 
especially  in  the  advanced  classes  of  the  high  school,  and  in  the 
college.  Some  teachers  favor  exclusively  the  so-called  recita- 
tion method ; others  believe  that  the  best  results  will  be  obtained 
by  explaining  the  subject  to  the  class  for  the  entire  period. 
This  method  most  frequently  results  in  a formal  lecture.  By 
means  of  an  examination  in  the  subject  at  the  end  of  the  term 
or  of  the  year,  or  by  occasional  tests  during  the  term,  the  teacher 
gains  information  as  to  the  standing  in  the  subject  of  the 
members  of  the  class.  But  neither  of  these  methods  of  conduct- 
ing the  work  with  a class  does  justice  to  the  pupil.  It  is  the 
duty  of  the  teacher  to  find  out  the  state  of  knowledge  of  the 
student  from  day  to  day.  In  this  way  the  teacher  will  see  in 
what  particular  the  student’s  understanding  of  the  work  is 
wanting.  By  giving  the  necessary  explanation  he  will  make  the 
ideas  and  the  work  clear  to  the  student.  This  method  with  some 
modifications  ought  to  be  used  even  in  the  work  of  the  graduate 
and  the  professional  school. 

We  can  never  be  sure  whether  or' not  we  know  a thing,  or 
understand  an  idea,  unless  we  come  to  practice  it,  or  to  express 
it  or  to  apply  it.  One  of  the  means  of  acquiring  clear  ideas  is  to 
speak  the  ideas  out  or  to  write  them  out.  We  must  have  the 
student  talk  the  ideas  over  with  us.  In  this  way  only  will  he 
find  out  whether  or  not  they  are  perfectly  clear  to  him.  Human 
nature  is  such  that  we  are  apt  to  do  promptly  what  we  must 
give  an  immediate  account  of,  and  postpone  such  work  for 
which  we  are  not  at  once  accountable. 

While  I thoroughly  believe  that  a person  cannot  do  as  good 
work  at  one  time  as  he  can  at  another,  and  that  we  must  in  a 
certain  sense  be  disposed  to  do  our  work,  yet  practical  life  does 
not  tolerate  such  tendencies  and  variances  of  the  mind.  In  life, 
we  have  to  do  our  day’s  work,  whatever  our  personal  inclina- 
tions may  be.  We  ought  therefore  to  train  the  youth  to  do  his 
work  from  day  to  day,  and  not  to  give  way  to  slight  indisposi- 
tions and  moods,  unless  the  mind  and  body  are  in  such  a state 
that  it  would  be  wiser  to  submit  than  to  try  to  conquer  them. 

Besides  getting  clear  ideas  of  the  conceptions  and  laws  of 
mathematics,  the  student  must  also  acquire  facility  in  perform- 
ing its  various  operations.  The  acquiring  of  this  facility  is  an 


62 


THE  MATHEMATICS  TEACHER. 


excellent  discipline,  and  is  necessary  if  the  student  is  to  continue 
the  study  of  mathematics.  Part  of  this  facility  the  student  must 
acquire  by  means  of  home  work.  He  will  have  to  work  at  the 
examples  assigned  to  him,  until  he  has  entire  mastery  of  all  of 
the  operations  involved  and  can  with  little  effort  carry  them 
through  to  correct  and  accurate  results. 

The  teacher  must  indicate  the  mistakes  which  the  student  has 
made  in  such  work  and  supplement  the  work  of  the  student  in 
such  a manner  as  to  make  it  entirely  clear  to  him.  The  teacher 
must  continually  encourage  the  student  to  ask  pertinent  ques- 
tions and  not  rest  content  until  he  has  entirely  mastered  each 
point  of  the  lesson. 

Very  often  a student  refrains  from  asking  a question  because 
of  a somewhat  indefinite  idea  which  seems  to  exist  in  the  mind 
of  some  of  them,  that  they  have,  so  to  speak,  no  claim  on  the 
services  of  the  teacher,  but  that  these  services  are  rendered,  in 
a sense,  as  a matter  of  favor.  If  such  an  opinion  is  prevalent, 
the  teacher  must  make  it  clear  to  the  student  that  it  is  his  duty 
to  give  the  student  all  the  information  and  help  he  may  need, 
and  to  answer  such  questions  as  will  make  the  ideas  presented 
entirely  clear  to  him.  If  the  teacher  is  strong  in  character  and 
in  mind — and  only  such  ought  to  be  employed — the  clear  under- 
standing which  the  students  will  have  of  the  teacher’s  duties 
and  functions  will  not  lessen  their  respect  for  him  and  not 
minimize  his  importance. 

Since  the  acquisition  of  knowledge  is,  as  we  have  stated, 
a matter  of  evolution,  we  must  do  our  work  from  day  to  day. 
We  must  digest  and  assimilate  the  ideas  we  learn  one  by  one, 
if  we  are  to  acquire  clear  conceptions  of  them,  and  if  they  are 
to  help  our  mental  development  and  increase  our  mental  power. 

In  most  of  the  foreign  universities  attendance  at  classes  is 
not  compulsory.  The  student  is  not  held  accountable  for  his 
work  from  day  to  day.  The  work  in  the  classroom  consists  of 
more  or  less  formal  lectures  delivered  by  the  teacher. 

It  is  })rcposterous  to  think  that  a young  man  of  eighteen  or 
nineteen,  with  very  little  experience  of  the  world  and  the  things 
in  it,  and  often  with  little  maturity  of  mind,  should  consider 
it  an  affront  if  he  made  to  attend  lectures  and  submit  to 
being  (jiiestioned,  unless  he  does  it  from  his  own  volition. 
To  he  com])elled  to  attend  lectures  and  recite,  would,  in  the 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  63 

opinion  of  these  students  be  a violation  of  their  so  much  cher- 
ished “ Akademische  Freiheit  ” (academic  freedom). 

In  some  of  these  universities  the  student,  whatever  his  age, 
considers  himself  as  belonging  to  an  exclusive  and  privileged 
class,  and  I cannot  help  but  feel  that  he  is  often  encouraged  in 
this  belief  by  the  authorities  of  the  university.  If  he  commits 
an  offence  which  would  make  him  liable  to  punishment,  in  ac- 
cordance with  the  law  as  applied  to  any  other  person,  he  is  in 
some  of  the  university  towns,  not  judged  by  the  properly  con- 
stituted courts,  but  is  left  by  them  to  be  judged  by  the  officers  of 
the  university.  The  punishment  inflicted  by  these  authorities 
generally  consists  in  incarceration  in  the  university  prison 
(Karcer).  The  jolly  hours  spent  by  the  student  in  this 
“ Karcer  ” have  been  described  in  prose  and  verse. 

The  general  conviction  in  this  country  that  all  men  are  equal 
before  the  law,  and  the  consequent  respect  for  the  law,  has  a 
beneficial  effect  on  the  morals  of  the  people. 

In  spite  of  the  fact  that  in  nearly  every  country  in  Europe, 
an  accurate  record  is  kept  of  the  occupation  of  every  inhabitant, 
of  his  previous  residence,  etc.,  I feel  that  the  personal  safety 
and  the  moral  status  here,  where  all  these  regulations  and  pre- 
cautions do  not  exist,  are  much  greater  than  in  most  European 
countries. 

The  system  of  education  as  it  is  carried  on  in  some  of  the 
foreign  universities  is  bound  to  affect  the  intellectual  and  moral 
status  of  the  entire  nation.  If  we  can  judge  the  signs  of  the 
times  and  predict  the  future,  the  center  of  gravity  of  intellectual 
powers  is  bound  to  shift  from  such  nations  to  those  where  edu- 
cation is  conducted  in  a more  effective  manner. 

While  I am  not  able  to  support  my  arguments  by  statistics,  it 
is  my  own  observation  and  experience  that  the  number  of  stu- 
dents who  leave  some  of  the  foreign  universities  without  finish- 
ing their  studies  is  growing  larger  and  larger. 

Such  men,  as  a rule,  become  a burden  to  society,  and  are  a 
curse  to  civilization.  Having  been  accustomed  to  the  “ Aka- 
demische  Freiheit,”  many  of  them  are  not  fit  to  take  employment 
and  to  be  subservient.  They  augment  the  ranks  of  those  who 
are  dissatisfied  with  the  world  and  the  things  in  it.  Some  of 
them  become  “ walking  delegates,”  or — I rather  hesitate  to  ex- 
press it — missionaries  to  foreign  countries. 


64 


THE  MATHEMATICS  TEACHER. 


This  shall  not  he  construed  as  showing  on  my  ])art  a lack  of 
appreciation  of  the  self-sacrificing  men  and  women  who  are 
willing  to  go  to  far-ofif  lands  and  administer  to  those  who  need 
their  help.  There  are  men  and  women  who  have  felt  the  calling 
to  be  missionaries,  and  many  already  while  in  youth  have  pre- 
pared themselves  for  their  chosen  life  work. 

The  teachers  in  some  of  the  foreign  universities  consider 
research  in  their  special  branch  of  learning,  as  their  principal 
duty,  and  the  instruction  of  the  students  a matter  of  secondary 
importance. 

They  pursue  this  course  not  as  a result  of  any  lack  of  a sense 
of  duty,  but  as  a matter  of  custom  and  tradition,  dating  from 
the  time  when  only  a few,  and  those  the  ablest,  attended  a 
university.  These  students  came  to  the  university  more  for  the 
sake  of  getting  the  stimulus  and  the  inspiration  which  comes 
from  contact  with  great  minds,  than  for  the  purpose  of  instruc- 
tion. But  a great  injustice  is  done  to  the  student  who  attends  a 
university  in  order  to  acquire  knowledge  and  obtain  informa- 
tion, if  he  does  not  receive  instruction  in  his  studies  from  those 
whose  duty  it  is  to  give  it. 

To  many  of  these  investigators  mankind  owes  a debt  of  grati- 
tude for  the  unselfish  manner  in  which  they  have  devoted  them- 
selves to  the  advancement  of  science.  They  have  by  their  labors, 
not  only  increased  knowledge,  but  also  have  added  to  the  com- 
forts and  conveniences  of  life,  and  often  their  discoveries  and 
inventions  have  been  the  means  of  producing  wealth.  The  only 
reward  for  their  efiforts  is  the  pleasure  and  satisfaction  which 
comes  from  searching  for  truth  and  finding  it. 

The  man  who  makes  the  discovery  or  the  invention,  especially 
if  it  is  along  scientific  lines,  does  not,  as  a rule,  reap  any 
material  benefit  from  his  abilities  and  labor.  One  of  the  most 
touching  incidents  I know  of  is  that  of  Heinrich  Hertz,  the  in- 
ventor of  wireless  telegraphy.  He  sacrificed  his  life  for  science, 
and  overwork  led  to  a premature  death.  Science  is  enriched  by 
his  labors.  The  comfort  and  convenience  which  his  invention 
has  added  to  life  is  hard  to  estimate.  It  is  pathetic  to  think  that 
his  name  is  not  even  connected  with  the  invention,  and  that  he 
is  not  even  given  j)opular  credit  for  his  labors. 

It  is  true  that  there  are  men  who,  while  carrying  on  research 
in  their  chosen  science,  are  still  able  to  descend  to  the  level  of 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  65 

the  beginner,  and  maintain  an  intellectual  companionship  with 
even  the  youngest  student.  Such  teachers,  I fear,  are  the  ex- 
ception. I feel  that  the  majority  of  the  teachers  when  engaged 
in  the  all-absorbing  work  of  scientific  research,  are  not  apt  to 
be  in  sympathy  with  the  needs  of  the  beginner  in  the  subject. 

Hegel  is  said  to  have  calmly  finished  his  “ Phaenomenologie 
des  Geistes  ” at  Jena,  on  October  14,  1806,  not  knowing  anything 
whatever  of  the  battle  that  was  raging  around  him.  The  carry- 
ing on  of  original  investigation,  especially  in  the  pure  sciences, 
requires  the  closest  concentration  of  our  mental  powers  which 
very  often  absorbs  our  physical  strength  and  endangers  our 
health,  and  may  thus  unfit  us  for  the  difficult  task  of  teaching  the 
young.  I also  quote  in  this  connection  from  a letter  of  the  great 
Jacobi : 

“ It  is  a wearisome  work  which  I have  done,  wearisome  work 
in  which  I am  engaged.  It  is  neither  diligence  nor  memory 
which  leads  here  to  the  goal,  for  they  are  here  merely  the  sub- 
servient ministers  of  the  moving  pure  thought.  But  tenacious, 
brain-splitting  pondering  (Nachdenken)  requires  more  strength 
than  the  most  persistent  diligence.  If  therefore,  by  the  con- 
stant practice  of  this  pondering  I have  acquired  some  power 
in  it,  let  it  not  be  thought  that  it  has  been  made  easy  for  me, 
perhaps  by  a fortunate  gift  of  nature.  Wearisome,  wearisome 
work  I have  had  to  undergo,  and  the  anxiety  of  the  pondering 
has  often  seriously  endangered  my  health.  The  consciousness, 
however,  of  power  attained,  forms  the  best  reward  of  the  work, 
and  is  also  an  encouragement  to  continue  and  not  to  relax.” 

The  young  ought  to  be  taught  not  only  to  husband  their  physi- 
cal strength,  but  to  be  economical  with  their  time  as  well.  It 
is  most  important  that  the  student,  whether  in  preparing  his 
lessons  or  in  his  reading,  learn  to  accomplish  the  best  results 
with  the  least  expenditure  of  time.  In  other  words,  he  must 
learn  how  to  study  and  how  to  work. 

The  method  of  the  spiral  system  which  has  been  successfully 
used  in  the  teaching  of  arithmetic  and  which  ought  to  be  applied 
to  the  teaching  of  each  of  the  branches  of  mathematics,  ought 
also  to  be  followed  in  reading  and  studying  any  subject,  especi- 
ally the  more  profound  ones.  It  would  be  a waste  of  time,  and 
with  some  people,  a source  of  discouragement,  to  attempt  to 
read  a difficult  work,  say  on  philosophy,  and  endeavor  to  have 


66 


THE  MATHEMATICS  TEACHER. 


a thorough  understanding  of  it  as  we  go  along.  A good  plan  is 
to  read  the  whole  book,  or  a part  of  it  carefully,  without  at- 
tempting to  thoroughly  understand  its  most  difficult  ])assagcs. 
The  second  reading  of  the  book  will  clarify  some  of  these  pas- 
sages which  are  not  clear  at  the  first  reading.  Repeated  readings 
will  make  the  book  clearer  and  clearer  until  a thorough  under- 
standing of  the  whole  work  is  reached. 

In  mathematics,  the  student  must  go  over  the  lesson  carefully, 
concentrate  his  mind  and  his  attention  on  it,  without  using  an 
undue  amount  of  time  on  the  more  difficult  parts.  After  going 
over  the  lesson  a second  time,  he  will  find  that  the  ideas  are 
much  clearer  to  him.  He  will  have  to  go  over  the  lesson  again 
and  again  until  he  has  a thorough  mastery  of  it. 

The  student  must  learn  to  concentrate  all  of  his  thoughts  on 
a piece  of  work  in  hand,  and  must  not  allow  his  attention  to  be 
distracted  by  exterior  influences.  He  ought  to  be  trained  to 
think  and  to  meditate.  Meditation  is  a great  help  towards 
clarifying  our  ideas  on  subjects  we  have  learned,  or  we  have 
heard,  or  read  or  seen.  Meditation  develops  and  strengthens 
the  power  and  the  habit  of  thinking.  In  the  rush  of  life,  we  do 
not  meditate  enough.  The  youth  must  be  constantly  reminded 
and  persistently  urged  to  make  the  best  use  of  his  opportunities 
and  of  his  time. 

If  I am  permitted  to  be  again  personal — and  I feel  that  we 
certainly  can  speak  more  authoritatively  about  our  own  experi- 
ence than  about  the  experience  of  others — there  is  nothing  that 
fills  me  with  so  much  regret  as  to  think  of  the  days  and  the 
hours  of  which  I have  not  made  proper  use.  We  feel  the  regret 
the  more,  as  we  grow  older.  O ! for  the  hours  wasted  which 
never  return ; for  the  opportunities  lost  which  never  come  again. 

I do  not,  however,  wish  it  to  be  understood  that  the  young 
ought  to  be  expected  to  employ  their  time  and  energy  only  in 
study  and  in  such  a manner  as  will  be  of  immediate  use  and 
direct  benefit  to  them. 

I'ime  si)ent  in  well  directed  out-door  exercises,  time  spent  in 
reading  a good  book,  or  in  the  company  of  a stimulating  friend, 
in  nursing  good  comradeship;  time  spent  in  listening  to  good 
music,  in  seeing  a good  l)lay,  or  in  watching  manly  sport,  is  time 
well  sj)ent. 

It  would  ai)pear  that  the  teacher  in  the  college  or  in  the 


MODERN'  TENDENCIES  IN  TEACHING  MATHEMATICS.  6/ 

university  is,  so  to  speak,  the  court  of  the  last  resort,  and  is  the 
only  judge  of  his  efforts,  yet  it  is  far  from  being  so. 

While  the  work  of  the  secondary  and  elementary  schools  has 
been  lately  subject  to  a great  deal  of  criticism,  the  work  of  the 
college  has  been  attacked  by  men  of  affairs  on  whose  sympathy 
with  the  college  its  maintenance  depends,  and  its  great  mission 
is  made  possible. 

These  attacks  have  been  found  to  be  worthy  of  comment, 
even  by  some  of  the  most  conservative  college  presidents. 
While  the  work  in  the  college  depends  on  the  preparation  which 
the  pupil  receives  in  the  elementary  and  secondary  schools,  yet 
the  college  is  judged  irrespective  of  this  preparation,  and  only 
by  what  the  finished  product — the  graduate  of  the  college — is 
able  to  do  and  by  what  he  represents. 

Those  who  have  been  successful  in  life,  in  spite  of  their  lack 
of  college  education,  or,  as  in  some  cases  without  any  school 
education  because  of  their  superior  qualities  of  character  and 
abilities  of  mind,  and  perhaps  also  because  they  were  aided  by 
favorable  conditions,  are  apt  to  be  rather  skeptical  of  the  bene- 
fits to  be  derived  from  higher  education.  There  are  some  who 
believe  that  a higher  education  even  unfits  a man  or  woman 
for  the  work  of  life.  I cannot  help  thinking  that  such  criticisms 
are  merely  the  result  of  experience  with  those  college  men,  who 
have  failed  to  make  proper  use  of  the  opportunities  and  facilities 
offered  by  the  institutions  which  they  have  attended.  A person 
may  go  through  college,  and  yet  not  be  educated. 

The  amount  of  ground  covered  in  a subject  does  not  deter- 
mine the  mental  status  of  the  learner.  The  inventory  of  the 
mental  powers  of  a person  cannot  be  measured  by  the  number 
of  books  read,  by  the  number  of  topics  studied,  by  the  number 
of  facts  learned,  but  only  by  the  number  of  clear  ideas  acquired, 
and  to  what  extent  he  is  able  to  supply  them  to  his  own  moral 
and  physical  betterment,  and  to  the  welfare  of  his  fellowmen. 
Such  criticisms  will  be  silenced  when  the  young  realize  the  true 
purpose  for  which  they  go  to  school  and  college,  when  they 
avail  themselves  of  the  opportunities,  and  make  proper  use  of 
the  facilities  offered  for  their  physical,  moral  and  intellectual 
development. 

True  education  ought  to  teach  the  youth  that  all  work  is 
ennobling  if  it  is  done  well  and  if  it  helps  towards  obtaining  the 


68 


THE  MATHEMATICS  TEACHER. 


necessary  comforts  and  conveniences  of  life.  That  experience 
gives  facility,  and  without  facility  the  most  effective  results 
cannot  be  obtained.  To  do  our  duty  should  be  our  highest 
ambition.  The  greater  our  opportunities,  the  greater  our  re- 
sponsibilities. A man’s  worth  ought  to  be  measured  by  his 
character  only,  and  the  man  who  has  been  fortunate  enough  to 
enjoy  the  advantages  of  education,  ought  to  become  a missionary 
for  all  that  is  good  and  noble.  He  should  administer  his  ideas 
by  precept  and  example  to  those  who  have  not  enjoyed  such 
advantages  as  he  has. 

Idleness  and  vice  go  hand  in  hand,  but  neither  ever  remains 
unpunished.  Reward  and  punishment  do  not  in  general  follow 
our  actions  immediately,  but  will  surely  come  some  day,  per- 
haps not  until  the  lapse  of  generations.  Leisure  is  only  to  be 
enjoyed  by  those  who  have  done  their  life’s  work,  and  who 
know  how  to  use  it. 

True  arguments  must  not  be  supported  by  quotations,  but 
must  be  substantiated  by  reasoning,  or  by  results  which  confirm 
the  experience  of  mankind. 

Education  must  teach  the  student  true  modesty.  He  must 
learn  that  our  knowledge  is  little  as  compared  with  what  we  still 
have  to  learn ; that  we  are  but  passing  trifles  when  compared 
with  the  eternity  of  matter  and  the  immensity  of  the  universe ; 
that  unless  we  contribute  something  towards  helping  the  world 
to  become  better  than  we  find  it,  we  have  lived  in  vain. 

It  is  the  wish  and  the  hope  of  parents  that  their  children  be 
in  every  way  better  than  they  are.  It  must  be  our  purpose  to 
leave  the  world  better  than  we  have  found  it.  In  this,  and  only 
in  this  way,  is  the  progress  of  the  world  morally,  intellectually 
and  socially  possible. 

While  a knowledge  of  the  history  of  mathematics  and  of  all 
other  kinds  of  information  as  to  the  development  and  the  philo- 
sophical aspects  of  the  subject  will  broaden  the  teacher  and  make 
him  better  fit  for  his  work,  yet  all  this  will  be  of  little  avail  if 
he  has  not  a thorough  knowledge  of  the  subject  he  is  teaching, 
])erfect  facility  in  solving  any  of  the  examples  and  problems  in 
connection  with  the  work,  and  such  a mastery  and  resourceful- 
ness in  the  subject  which  is  above  the  hook  and  syllabus  and  as 
will  enable  him  to  adapt  the  instruction  to  the  needs  of  the 
students  of  different  ability,  and  different  environments. 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS.  69 

Better  text-books,  a change  in  the  number  and  the  kind  of 
subject  taught,  a better  distribution  of  the  hours  for  work  and 
the  hours  for  recreation,  both  for  pupils  and  teachers,  a change 
in  the  ratio  between  the  number  of  pupils  and  the  number  of 
teachers,  will  help  and  help  a great  deal  if  the  teacher  has  the 
highest  qualities  of  character  and  mind,  and  is  able  to  impress 
his  pupils  with  them,  and  use  them  to  make  the  pupils  follow  his 
teachings  and  advice  in  the  classroom  and  out.  The  most 
efficient  supervision,  the  wisest  direction  by  school  officers,  all 
advances  in  the  physical  equipment  of  the  school  will  be  of  little 
avail,  unless  the  government  of  the  school  is  able  to  attract  and 
retain  men  and  women  of  the  highest  qualities  of  character  and 
mind,  well  equipped  for  their  work.  The  plans  of  the  architect 
are  of  little  use,  unless  the  builder  and  sculptor  are  able  to 
execute  them  and  to  give  form  to  the  plans. 

The  teacher  must  understand  the  peculiarities  of  the  student 
and  their  difficulties,  he  must  be  able  to  descend  to  the  level  of 
the  student  and  to  enter  into  an  intellectual  companionship  even 
with  the  beginner  in  a subject,  and  also  with  those  of  little 
ability.  If  the  teacher  will  do  this,  then  such  charges  as  are 
often  made  by  students,  that  while  the  teacher  seems  to  have  a 
knowledge  of  the  subject,  he  does  not  seem  able  to  impart  it  to 
his  students,  will  no  more  be  heard. 

The  mode  of  presenting  a subject,  the  manner  of  handling  a 
class  cannot  very  well  be  learned  from  books.  The  description 
of  an  event  is  no  match  to  actually  seeing  it  happening.  The 
reading  of  instructions  is  only  a weak  substitute  for  the  “ living 
word,”  and  seeing  these  instructions  actually  carried  out.  No 
one  should  have  the  right  to  teach  unless  he  attends  for  a certain 
period  of  time  the  classes  in  actual  progress  of  men  and  women 
who  have  reached  a high  proficiency  of  imparting  knowledge  and 
conducting  classes.  The  candidate  must  have  studied  and  be 
proficient  in  the  subject  of  which  such  a class  is  treating,  in 
order  that  he  may  be  able  to  concentrate  his  attention  and  efforts 
on  the  mode  of  presentation,  and  the  manner  in  which  the  class 
is  conducted. 

What  we  need  more  than  anything  else  is  to  attract  men  and 
women  to  our  profession  who  not  only  have  perfect  mastery 
of  the  subject  they  are  teaching,  and  the  ability  to  impart  the 
knowledge  to  their  pupils,  but  also  who  have  the  personality,  the 


70 


THE  MATHEMATICS  TEACHER. 


enthusiasm  for  their  high  calling,  and  all  those  qualifications 
of  character  and  mind  to  make  them  an  inspiration  and  a stimu- 
lus to  their  students.  There  is  no  occupation  in  which  (jualities 
of  sincerity  and  frankness  and  the  most  extreme  straightfor- 
wardness are  more  important  than  in  teaching.  If  the  teacher 
is  to  accomplish  his  great  task,  he  must  have  the  complete  con- 
fidence of  his  pupils.  The  teacher  exerts  the  most  lasting  in- 
fluence during  the  formative  period  of  the  child’s  character.  The 
character  of  the  teacher,  his  habits,  his  entire  life  must  be  models 
after  which  the  student  can  most  safely  pattern  his  own  life. 

While  the  profession  of  the  teacher  is  in  a sense  rather  re- 
spected, yet  on  the  whole,  the  teacher  does  not  enjoy  the  appre- 
ciation which  is  accorded  to  members  of  other  professions.  I 
feel  that  the  public  at  large,  and  especially  men  of  affairs,  are 
inclined  to  consider  the  teacher  whose  actions  and  activities 
extend  only  to  the  young,  as  not  possessing  such  qualities  as 
the  men  who  are  in  daily  contact  with  the  practical  world. 

If  the  teacher  is  to  discharge  his  important  duties  of  develop- 
ing the  character  and  the  mind  of  the  young,  and  thus  effecting 
the  amelioration  of  the  race,  he  must  be  well  informed  about 
the  world  and  the  things  in  it,  about  everything  that  is  good  and 
noble  and  everything  that  needs  to  be  improved  and  remedied. 

The  teacher  with  the  opportunities  for  self-culture  and  medi- 
tation, ought  to  be  best  equipped  to  take  a leading  part  in  every 
movement  that  will  make  for  moral  and  intellectual  betterment. 

While  the  profession  of  the  teacher  has  the  attraction,  that 
it  gives  more  leisure  than  is  enjoyed  by  the  members  of  most 
of  the  other  professions,  it  does  not,  on  the  other  hand,  offer 
such  possibilities  as  other  professions  hold  out  to  those  who 
enter  them.  The  teacher  on  entering  his  profession  knows 
pretty  well  what  limit  he  is  able  to  reach;  he  cannot,  on  the 
whole,  aspire  to  reach  such  a goal  as  the  average  American  boy 
may  set  for  himself  when  he  enters  other  professions  or  business. 

The  tendency  has  prevailed  at  all  times,  and  is  stronger  now 
than  ever  before  to  offer  in  any  of  the  activities  of  life,  high 
rewards  for  superior  qualifications  and  high  efficiency  of  service. 
The  same  tendencies  ought  to  apply  to  the  teaching  profession 
also.  Teaching  is  an  art,  and  a high  and  a noble  art.  It  is  a 
(iofl-given  art  to  cut  a statue  out  of  a piece  of  marble,  which 
after  all  is  only  subject  to  the  chisel  of  the  sculptor.  What 


MODERN  TENDENCIES  IN  TEACHING  MATHEMATICS. 


71 


nobler  art  it  is  to  mould  the  character  and  the  mind  of  the  child 
— with  its  environments,  home  influence  and  temperament — so 
as  to  make  it,  morally,  mentally  and  physically  as  nearly  as 
possible,  a perfect  man  or  woman. 

There  is  a difference  in  the  degree  of  ability  necessary  to  solve 
the  various  problems  which  the  follower  of  most  of  the  profes- 
sions meets  in  his  practice.  It  is  not  as  difficult  to  diagnose 
and  cure  some  diseases  as  it  to  conquer  others.  Some  engineer- 
ing problems  require  for  their  solution  a high  degree  of  ability 
and  often  great  ingenuity,  while  a moderate  degree  of  ability 
is  necessary  to  solve  others.  The  same  is  true  of  the  problems 
in  the  other  professions.  As  far  as  the  teaching  profession 
is  concerned,  it  takes  as  high  a degree  of  ability,  of  character 
and  mind,  and  of  all  other  high  qualities,  to  mould  the  character 
and  mind  of  the  child  attending  the  lowest  grades  of  the  ele- 
mentary schools  as  it  does  to  influence  the  student  in  the  highest 
classes  of  the  college.  In  all  other  professions  and  occupations 
there  is  room  for  persons  with  moderate  ability.  If  they  per- 
form their  duties  conscientiously,  their  services  are  as  neces- 
sary and  important  as  are  the  services  of  those  who  are  engaged 
in  the  more  difficult  problems  of  their  profession.  In  the  teach- 
ing profession  there  is  room  only  for  those  of  highest  ability  and 
superior  qualities  of  character,  mind  and  heart,  if  education  is 
to  accomplish  its  great  mission,  the  amelioration  of  the  race. 

The  public  provides,  on  the  whole,  liberally  for  the  necessities 
of  the  community.  If  it  could  be  made  to  see  the  great  influence 
and  the  mission  of  education,  then  the  respect  for  education 
and  for  the  teacher  will  rise,  and  he  will  be  accorded  in  every 
way  the  credit  which  he  deserves. 

University  of  Pennsylvania, 

Philadelphia,  Pa. 


UNIVERSfTY  OF  ILLINOIS-URBANA 


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